challenges / van-der-waerden-lower
Van der Waerden numbers, lower bounds by explicit colourings
An r-colouring of {1..N} with no monochromatic k-term arithmetic progression certifies W(r,k) > N. Thirteen open (r,k) pairs from (2,7) to (6,4), scored against the longest known certificates (Rabung, Herwig-Heule-van Lambalgen-van Maaren, Rabung-Lotts, Heule).
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# Van der Waerden numbers: lower bounds by explicit colourings
## Goal
The van der Waerden number `W(r, k)` is the least `n` such that every `r`-colouring of `{1, ..., n}`
contains a monochromatic `k`-term arithmetic progression `a, a+d, ..., a+(k-1)d` (`d ≥ 1`). Only seven
non-trivial values are known (`W(2,3..6) = 9, 35, 178, 1132`, `W(3,3) = 27`, `W(3,4) = 293`,
`W(4,3) = 76`). For every other pair the only handle is a *certificate*: an `r`-colouring of
`{1, ..., N}` with no monochromatic `k`-AP proves `W(r, k) > N`. This pack scores such certificates
for thirteen open pairs against the longest ones known.
`colouring.py` exposes
colouring(r: int, k: int, time_budget: float, seed: int) -> list[int]
…